Optimal. Leaf size=23 \[ -\frac {x}{c}-\frac {2 a \log (a-b x)}{b c} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45}
\begin {gather*} -\frac {2 a \log (a-b x)}{b c}-\frac {x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {a+b x}{a c-b c x} \, dx &=\int \left (-\frac {1}{c}+\frac {2 a}{c (a-b x)}\right ) \, dx\\ &=-\frac {x}{c}-\frac {2 a \log (a-b x)}{b c}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {x}{c}-\frac {2 a \log (a-b x)}{b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 22, normalized size = 0.96
method | result | size |
default | \(\frac {-x -\frac {2 a \ln \left (-b x +a \right )}{b}}{c}\) | \(22\) |
norman | \(-\frac {x}{c}-\frac {2 a \ln \left (-b x +a \right )}{b c}\) | \(24\) |
risch | \(-\frac {x}{c}-\frac {2 a \ln \left (-b x +a \right )}{b c}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x}{c} - \frac {2 \, a \log \left (b x - a\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.06, size = 23, normalized size = 1.00 \begin {gather*} -\frac {b x + 2 \, a \log \left (b x - a\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 0.74 \begin {gather*} - \frac {2 a \log {\left (- a + b x \right )}}{b c} - \frac {x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.19, size = 25, normalized size = 1.09 \begin {gather*} -\frac {x}{c} - \frac {2 \, a \log \left ({\left | b x - a \right |}\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 23, normalized size = 1.00 \begin {gather*} -\frac {b\,x+2\,a\,\ln \left (b\,x-a\right )}{b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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